Theorem eulerthlem1 12917

Description: Lemma for eulerth 12923. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
eulerthlem1.1	|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
eulerthlem1.2	|- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
eulerthlem1.3	|- T = ( 1 ... ( phi ` N ) )
eulerthlem1.4	|- ( ph -> F : T -1-1-onto-> S )
eulerthlem1.5	|- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )

Assertion

Ref	Expression
eulerthlem1	|- ( ph -> G : T --> S )

Distinct variable groups:   x, y, A   x, F, y   x, G, y   x, N, y   x, S   ph, x, y   x, T, y

Allowed substitution hint:   S( y)

Proof of Theorem eulerthlem1

Step	Hyp	Ref	Expression
1		eulerthlem1.1	. . . . . . 7 |- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
2	1	simp2d 1037	. . . . . 6 |- ( ph -> A e. ZZ )
3	2	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> A e. ZZ )
4		eulerthlem1.4	. . . . . . . . . 10 |- ( ph -> F : T -1-1-onto-> S )
5		f1of 5613	. . . . . . . . . 10 |- ( F : T -1-1-onto-> S -> F : T --> S )
6	4, 5	syl 14	. . . . . . . . 9 |- ( ph -> F : T --> S )
7	6	ffvelcdmda 5811	. . . . . . . 8 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. S )
8		oveq1 6056	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y gcd N ) = ( ( F ` x ) gcd N ) )
9	8	eqeq1d 2241	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( y gcd N ) = 1 <-> ( ( F ` x ) gcd N ) = 1 ) )
10		eulerthlem1.2	. . . . . . . . 9 |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
11	9, 10	elrab2 2975	. . . . . . . 8 |- ( ( F ` x ) e. S <-> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
12	7, 11	sylib 122	. . . . . . 7 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
13	12	simpld 112	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ( 0..^ N ) )
14		elfzoelz 10477	. . . . . 6 |- ( ( F ` x ) e. ( 0..^ N ) -> ( F ` x ) e. ZZ )
15	13, 14	syl 14	. . . . 5 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ZZ )
16	3, 15	zmulcld 9702	. . . 4 |- ( ( ph /\ x e. T ) -> ( A x. ( F ` x ) ) e. ZZ )
17	1	simp1d 1036	. . . . 5 |- ( ph -> N e. NN )
18	17	adantr 276	. . . 4 |- ( ( ph /\ x e. T ) -> N e. NN )
19		zmodfzo 10705	. . . 4 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
20	16, 18, 19	syl2anc 411	. . 3 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
21		modgcd 12680	. . . . 5 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
22	16, 18, 21	syl2anc 411	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
23	17	nnzd 9695	. . . . . 6 |- ( ph -> N e. ZZ )
24	23	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> N e. ZZ )
25	16, 24	gcdcomd 12663	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) gcd N ) = ( N gcd ( A x. ( F ` x ) ) ) )
26	23, 2	gcdcomd 12663	. . . . . . 7 |- ( ph -> ( N gcd A ) = ( A gcd N ) )
27	1	simp3d 1038	. . . . . . 7 |- ( ph -> ( A gcd N ) = 1 )
28	26, 27	eqtrd 2265	. . . . . 6 |- ( ph -> ( N gcd A ) = 1 )
29	28	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd A ) = 1 )
30	24, 15	gcdcomd 12663	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = ( ( F ` x ) gcd N ) )
31	12	simprd 114	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) gcd N ) = 1 )
32	30, 31	eqtrd 2265	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = 1 )
33		rpmul 12788	. . . . . 6 |- ( ( N e. ZZ /\ A e. ZZ /\ ( F ` x ) e. ZZ ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
34	24, 3, 15, 33	syl3anc 1274	. . . . 5 |- ( ( ph /\ x e. T ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
35	29, 32, 34	mp2and 433	. . . 4 |- ( ( ph /\ x e. T ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 )
36	22, 25, 35	3eqtrd 2269	. . 3 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 )
37		oveq1 6056	. . . . 5 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( y gcd N ) = ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) )
38	37	eqeq1d 2241	. . . 4 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( ( y gcd N ) = 1 <-> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
39	38, 10	elrab2 2975	. . 3 |- ( ( ( A x. ( F ` x ) ) mod N ) e. S <-> ( ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) /\ ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
40	20, 36, 39	sylanbrc 417	. 2 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. S )
41		eulerthlem1.5	. 2 |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )
42	40, 41	fmptd 5830	1 |- ( ph -> G : T --> S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   |-> cmpt 4170   -->wf 5347   -1-1-onto->wf1o 5350   ` cfv 5351  (class class class)co 6049   0cc0 8123   1c1 8124   x. cmul 8128   NNcn 9233   ZZcz 9573   ...cfz 10338  ..^cfzo 10472   mod cmo 10680   gcd cgcd 12642   phicphi 12899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-dvds 12467  df-gcd 12643

This theorem is referenced by:  eulerthlemh  12921  eulerthlemth  12922